Polygon centroid and interior point

Model: Geodesic edges cut into 5 km pieces and mapped to a Lambert azimuthal equal-area plane on the authalic sphere of WGS 84, centered at the corners' mean (an exact equal-area map, Snyder 1987 ch. 24 and eq. 3-11 to 3-16); the centroid is the plane's area-weighted centroid, holes subtracted, mapped back. The interior point is the pole of inaccessibility on that plane (Agafonkin's polylabel, to 0.1% of the shape's narrow side), with its clearance measured as a geodesic distance

Accuracy: The centroid is exact for the equal-area plane, which is what area-weighting on a curved surface needs a choice of; for shapes of a few hundred kilometers it matches a local plane to well under 1 m. The interior point is within 0.1% of the shape's narrow side of the true pole of inaccessibility; for extreme slivers (thousands of times longer than wide) it is the best found within a fixed amount of work, always inside

When to use this: Use this to find the middle of an area — the centre of a parcel, a search zone, a coverage polygon, a district — when the answer has to sit correctly on the ellipsoid rather than being the average of the corner coordinates, which is wrong for anything but a small symmetric shape. Two points come back, and they are for different jobs: the centroid is the centre of mass, which is what a calculation wants, and the interior point is guaranteed to be inside and as far from the edges as it can be, which is what a label or a map pin wants.

Limitations: The centroid of a shape on a curved surface is not defined until you choose how to weight area, and this weights it on an equal-area map, which is the choice that makes the answer independent of how the shape is oriented. For a concave shape — a C, a horseshoe, a ring — the centre of mass falls outside the polygon, which is correct and not an error; the tool says so and hands back the interior point instead. The interior point is found to within 0.1% of the shape’s narrow side, so it is the best place for a pin rather than a uniquely defined coordinate, and for a sliver thousands of times longer than it is wide it is the best found within a fixed amount of work. Holes are subtracted; self-intersecting outlines should be repaired first.

Worked example: A C-shaped lot, whose centroid falls in the notch. Source: a C shape, whose centre of mass falls outside it: CENTROID_OUTSIDE is raised and the interior point returned. Checked against PROJ 9.3.0's ellipsoidal Lambert azimuthal equal-area with GEOS 3.11.4's planar centroid over twelve shapes, which agree with the tool to 23 mm and to 4.6e-9 of the area. It is golden test vector v010, and every build checks the tool still gives its answer within its tolerance.

Review: Not yet independently reviewed by a GIS professional.

Last verified: 2026-09-19, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.

Status: version 1.0.0, core 0.1.0. See this tool in the verification report.

Checked against: 24 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

• Map Projections: A Working Manual, Snyder, J. P., U.S. Geological Survey, Professional Paper 1395. Chapter 24 (Lambert azimuthal equal-area) and equations 3-11 to 3-16 (authalic latitude).